124 research outputs found
Mixed Map Labeling
Point feature map labeling is a geometric problem, in which a set of input
points must be labeled with a set of disjoint rectangles (the bounding boxes of
the label texts). Typically, labeling models either use internal labels, which
must touch their feature point, or external (boundary) labels, which are placed
on one of the four sides of the input points' bounding box and which are
connected to their feature points by crossing-free leader lines. In this paper
we study polynomial-time algorithms for maximizing the number of internal
labels in a mixed labeling model that combines internal and external labels.
The model requires that all leaders are parallel to a given orientation , whose value influences the geometric properties and hence the
running times of our algorithms.Comment: Full version for the paper accepted at CIAC 201
Agglomerative Clustering of Growing Squares
We study an agglomerative clustering problem motivated by interactive glyphs
in geo-visualization. Consider a set of disjoint square glyphs on an
interactive map. When the user zooms out, the glyphs grow in size relative to
the map, possibly with different speeds. When two glyphs intersect, we wish to
replace them by a new glyph that captures the information of the intersecting
glyphs.
We present a fully dynamic kinetic data structure that maintains a set of
disjoint growing squares. Our data structure uses
space, supports queries in worst case time, and updates in
amortized time. This leads to an time
algorithm to solve the agglomerative clustering problem. This is a significant
improvement over the current best time algorithms.Comment: 14 pages, 7 figure
Dynamic Data Structures for k-Nearest Neighbor Queries
Our aim is to develop dynamic data structures that support -nearest neighbors (-NN) queries for a set of point sites in the plane in time, where is some polylogarithmic function of . The key component is a general query algorithm that allows us to find the -NN spread over substructures simultaneously, thus reducing an term in the query time to . Combining this technique with the logarithmic method allows us to turn any static -NN data structure into a data structure supporting both efficient insertions and queries. For the fully dynamic case, this technique allows us to recover the deterministic, worst-case, query time for the Euclidean distance claimed before, while preserving the polylogarithmic update times. We adapt this data structure to also support fully dynamic \emph{geodesic} -NN queries among a set of sites in a simple polygon. For this purpose, we design a shallow cutting based, deletion-only -NN data structure. More generally, we obtain a dynamic planar -NN data structure for any type of distance functions for which we can build vertical shallow cuttings. We apply all of our methods in the plane for the Euclidean distance, the geodesic distance, and general, constant-complexity, algebraic distance functions
Homotopy Measures for Representative Trajectories
An important task in trajectory analysis is defining a meaningful representative for a cluster of similar trajectories. Formally defining and computing such a representative r is a challenging problem. We propose and discuss two new definitions, both of which use only the geometry of the input trajectories. The definitions are based on the homotopy area as a measure of similarity between two curves, which is a minimum area swept by all possible deformations of one curve into the other. In the first definition we wish to minimize the maximum homotopy area between r and any input trajectory, whereas in the second definition we wish to minimize the sum of the homotopy areas between r and the input trajectories. For both definitions computing an optimal representative is NP-hard. However, for the case of minimizing the sum of the homotopy areas, an optimal representative can be found efficiently in a natural class of restricted inputs, namely, when the arrangement of trajectories forms a directed acyclic graph
Kinetic Geodesic Voronoi Diagrams in a Simple Polygon
We study the geodesic Voronoi diagram of a set S of n linearly moving sites inside a static simple polygon P with m vertices. We identify all events where the structure of the Voronoi diagram changes, bound the number of such events, and then develop a kinetic data structure (KDS) that maintains the geodesic Voronoi diagram as the sites move. To this end, we first analyze how often a single bisector, defined by two sites, or a single Voronoi center, defined by three sites, can change. For both these structures we prove that the number of such changes is at most O(m³), and that this is tight in the worst case. Moreover, we develop compact, responsive, local, and efficient kinetic data structures for both structures. Our data structures use linear space and process a worst-case optimal number of events. Our bisector KDS handles each event in O(log m) time, and our Voronoi center handles each event in O(log² m) time. Both structures can be extended to efficiently support updating the movement of the sites as well. Using these data structures as building blocks we obtain a compact KDS for maintaining the full geodesic Voronoi diagram
Chromatic k-Nearest Neighbor Queries
Let be a set of colored points. We develop efficient data structures
that store and can answer chromatic -nearest neighbor (-NN) queries.
Such a query consists of a query point and a number , and asks for the
color that appears most frequently among the points in closest to .
Answering such queries efficiently is the key to obtain fast -NN
classifiers. Our main aim is to obtain query times that are independent of
while using near-linear space.
We show that this is possible using a combination of two data structures. The
first data structure allow us to compute a region containing exactly the
-nearest neighbors of a query point , and the second data structure can
then report the most frequent color in such a region. This leads to linear
space data structures with query times of for points in
, and with query times varying between
and , depending on the distance measure used, for
points in . Since these query times are still fairly large we
also consider approximations. If we are allowed to report a color that appears
at least times, where is the frequency of the most
frequent color, we obtain a query time of in and expected query
times ranging between and
in using near-linear
space (ignoring polylogarithmic factors).Comment: 37 pages, 9 figure
Развитие финансово-кредитной инфраструктуры национальной экономики
We introduce a variation of unit-distance graphs which we call emph clear unit-distance graphs. They require the pairwise distances of the representing points to be either exactly 1 or not close to 1. We discuss properties and applications of clear unit-distance graphs
Improved Dynamic Geodesic Nearest Neighbor Searching in a Simple Polygon
We present an efficient dynamic data structure that supports geodesic nearest neighbor queries for a set S of point sites in a static simple polygon P. Our data structure allows us to insert a new site in S, delete a site from S, and ask for the site in S closest to an arbitrary query point q in P. All distances are measured using the geodesic distance, that is, the length of the shortest path that is completely contained in P. Our data structure achieves polylogarithmic update and query times, and uses O(n log^3n log m + m) space, where n is the number of sites in S and m is the number of vertices in P. The crucial ingredient in our data structure is an implicit representation of a vertical shallow cutting of the geodesic distance functions. We show that such an implicit representation exists, and that we can compute it efficiently
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